A330782 The records for distance squared for step lengths between adjacent composite numbers in A332767, the visited composite numbers for a walk stepping to the closest unvisited composite number on the 2D square (Ulam) spiral.

1, 2, 8, 32, 40, 68, 98, 148, 162, 356, 450

Offset: 1

Author

Scott R. Shannon, Feb 23 2020

Keywords

• nonn
• walk
• more

Comments

The sequence A332767 gives the visited composite numbers for a walk on the 2D square (Ulam) spiral which starts at 1 and then steps to the square containing the closest unvisited composite number. This sequences lists the records for the square of the step distance between visited composite numbers for that walk. For a walk of 1 million steps the largest square distance is 450, approximately 21.1 units, which occurs between A332767(149464) = 64666, which has coordinates (-127,-22) relative to the starting 1-square, to A332767(149465) = 67774 with coordinates (-130,-43). See A332767 for an image of the walk. It is unknown if this is a finite or infinite sequence.

Examples

			The below table shows the details of the record step lengths of this sequence for the first 1 million steps. The coordinate is relative to the starting 1-square.
--------------------------------------------------------------------------------
    a(n)  | A332767 step # |   Start value & coord   |  End value & coord      |
--------------------------------------------------------------------------------
       1  |         1      |         1 (0,0)         |         4 (0,1)         |
       2  |         6      |        32 (2,3)         |        30 (3,2)         |
       8  |       154      |        74 (-3,-4)       |       158 (-5,-6)       |
      32  |      4501      |      5526 (-37,-12)     |      6782 (-41,-16)     |
      40  |     65877      |     48150 (110,79)      |     53558 (116,81)      |
      68  |     91787      |    126154 (178,-49)     |    137780 (186,-47)     |
      98  |    125472      |    145762 (-28,191)     |    156654 (-35,198)     |
     148  |    142733      |    105316 (-147,-162)   |    102746 (-135,-160)   |
     162  |    142741      |     92744 (-129,-152)   |     82106 (-120,-143)   |
     356  |    142869      |     67818 (-130,-87)    |     57792 (-120,-71)    |
     450  |    149464      |     64666 (-127,-22)    |     67774 (-130,-43)    |
		

Links

• Wikipedia, Ulam Spiral.

Crossrefs

Cf. A332767, A330979, A331027, A000040, A063826, A136626.